7-1 7.3 sum and difference identities cosine sum and...
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7-1
7.3 Sum and Difference Identities
Cosine Sum and Difference Identities: cos A B does NOT equal cos cos .A B
Cosine of a Sum or Difference
cos ____________________________________A B
cos ____________________________________A B
EXAMPLE 1 Finding Exact Cosine Function Values Find the exact value of each expression.
(a) cos15 (b) cos 75°
Sine of a Sum or Difference
sin ____________________________________A B
sin ____________________________________A B
Tangent of a Sum or Difference
tan A B
tan A B
EXAMPLE 3 Finding Exact Sine and Tangent Function Values
Find the exact value of each expression.
(a) sin 75° (b) tan 105°
(c) sin 40° cos 160° − cos 40° sin 160° (d) cos87 cos93 sin87 sin93
EXAMPLE 4 Writing Functions as Expressions Involving Functions of
Write each function as an expression involving functions of .
(a) cos 30 (b) tan 45
(c) sin 180
EXAMPLE 5 Finding Function Values and the Quadrant of A + B
Suppose that A and B are angles in standard position, with 4
sin , ,5 2
A A
and 5 3
cos , .13 2
B B
Find each of the following.
(a) sin (A + B) (b) tan (A + B)
7-3
Verifying an Identity
EXAMPLE 7 Verifying an Identity
Verify that the following equation is an identity.
sin cos cos6 3
7.4 Double-Angle and Half-Angle Identities
■ Double-Angle Identities ■ Verifying an Identity
Double-Angle Identities
cos 2A = sin 2A =
cos 2A =
cos 2A = tan 2A =
EXAMPLE 1 Finding Function Values of 2 Given Information about
Given 3
cos5
and sin 0, find sin2 , cos2 , and tan2 .
EXAMPLE 2 Finding Function Values of Given Information about 2
Find the values of the six trigonometric functions of if 4
cos25
and 90 180 .
7-5
EXAMPLE 3 Simplifying Expressions Using Double-Angle Identities
Simplify each expression.
(a) 2 2cos 7 sin 7x x (b) sin15 cos15
EXAMPLE 4 Deriving a Multiple-Angle Identity
Write sin 3x in terms of sin x.
Half-Angle Identities
In the following identities, the symbol _____________ indicates that the sign is chosen based on the function
under consideration and the _____________ of .2
A
cos (𝐴
2) = ___________ sin (
𝐴
2) = ___________ tan (
𝐴
2) = ___________
EXAMPLE 9 Using a Half-Angle Identity to Find an Exact Value
Find the exact value of tan 22.5° using the identity sin
tan .2 1 cos
A A
A
7.5 Inverse Circular Functions
■ Review of Inverse Functions ■ Inverse Sine, Cosine, Tangent Functions ■ Remaining Inverse Circular
Functions ■ Inverse Function Values
Review of Inverse Functions
If a function is defined so that each ____________ element is used __________________
__________________, then it is called a one-to-one function.
Do not confuse the −1 in 1f with a negative exponent. The symbol 1f x does NOT represent
1.
f x It
represents the ____________ ____________ of f.
Review of Inverse Functions
1. In a one-to-one function, each x-value corresponds to ____________ ____________
y-value and each y-value corresponds to ____________ ____________ x-value.
2. If a function f is one-to-one, then f has an ____________ ____________ 1.f
3. The domain of f is the ____________ of 1,f and the range of f is the ____________ of 1.f That is, if the
point (a, b) is on the graph of f, then ____________ is on the graph of 1.f
4. The graphs of f and 1f are ____________ of each other across the line y = x.
5. To find 1f x from ,f x follow these steps.
Step 1 Replace f x with y and interchange x and y.
Step 2 Solve for y.
Step 3 Replace y with 1 .f x
Inverse Sine Function
1siny x
or arcsiny x means that sin , for .2 2
x y y
We can think of 1
siny x or arcsiny x as
“y is the angle in the interval ,2 2
whose sine is x.”
EXAMPLE 1 Finding Inverse Sine Values
Find y in each equation.
(a) 1
arcsin2
y (b) 1sin 1y (c) 1sin 2y
Be certain that the number given for an inverse function value is in the range of the particular inverse
function being considered.
7-7
Domain: _______________ Range: _________
Inverse Cosine Function
1cosy x
or arccosy x means that cos , for 0 .x y y
Domain: _______________ Range: _________
EXAMPLE 2 Finding Inverse Cosine Values
Find y in each equation.
(a) y = arccos (-1) (b) 1 2
cos2
y
(c) y = arccos (1/2)
x y
x y
Inverse Tangent Function
1tany x
or arctany x means that tan , for .2 2
x y y
Domain: _______________ Range: _________
Remaining Inverse Circular Functions
Inverse Cotangent, Secant, and Cosecant Functions
1coty x
or arccoty x means that cot , for 0 .x y y
1secy x
or arcsecy x means that sec , for 0 , .2
x y y y
1cscy x
or arccscy x means that csc , for , 0.2 2
x y y y
x y
7-9
Inverse Function Domain
Range
Interval
Quadrants of the
Unit Circle
1siny x
1cosy x
1tany x
1coty x
1secy x
1cscy x
EXAMPLE 3 Finding Inverse Function Values (Degree-Measured Angles)
Find the degree measure of in the following.
(a) arctan1 (b) 1sec 2
Use the following to evaluate these inverse trigonometric functions on a calculator.
1sec x
can be evaluated as 1 1cos ;
x
1csc x
can be evaluated as 1 1sin ;
x
1cot x
can be evaluated as
1
1
1tan if 0
1180 tan if 0.
xx
xx
EXAMPLE 4 Finding Inverse Function Values with a Calculator
Use a calculator to give each value.
(a) Find y in radians if 1csc 3 .y (b) Find in degrees if arccot 0.3541 .
Be careful when using your calculator to evaluate the inverse cotangent of a negative quantity. To do this, we
must enter the inverse tangent of the _____________ of the negative quantity, which returns an angle in
quadrant _____________. Since inverse cotangent is _____________ in quadrant II, adjust your calculator
result by adding 180° or accordingly. Note that 1cot 0 .2
EXAMPLE 5 Finding Function Values Using Definitions of the Trigonometric Functions
Evaluate each expression without using a calculator.
(a) sin (sin−1 (0.7)) (b) sin−1 (sin (3π
4))
(c) 1 3sin tan
2
(d) 1 5tan cos
13
EXAMPLE 6 Finding Function Values Using Identities
Evaluate each expression without using a calculator.
(a) 1
cos arctan 3 arcsin3
(b)
2tan 2arcsin
5
7-11
7.6 Solving Trigonometric Equations
■ Solving by Linear Methods ■ Solving by Factoring ■ Solving by Quadratic Methods
■ Solving by Using Trigonometric Identities ■ Equations with Half-Angles
■ Equations with Multiple Angles ■ Applications
Solving by Linear Methods
EXAMPLE 1 Solving a Trigonometric Equation by Linear Methods
Solve the equation 2sin 1 0
(a) over the interval [0°, 360°), and (b) for all solutions.
Solving by Factoring
EXAMPLE 2 Solving a Trigonometric Equation by Factoring
Solve sin tan sin over the interval [0°, 360°).
Solving by Quadratic Methods
EXAMPLE 3 Solving a Trigonometric Equation by Factoring
Solve 2tan tan 2 0x x over the interval 0, 2 .
EXAMPLE 4 Solving a Trigonometric Equation Using the Quadratic Formula
Find all solutions of cot cot 3 1.x x Write the solution set.
7-13
Solving a Trigonometric Equation
1. Decide whether the equation is ______________ or ______________ in form, so that you can determine
the solution method.
2. If only one trigonometric function is present, ______________ ______________ ______________ for
that function.
3. If more than one trigonometric function is present, rearrange the equation so that one side equals
______________. Then try to ______________and set each ______________equal to 0 to solve.
4. If the equation is quadratic in form, but not factorable, use the ______________ ______________. Check
that solutions are in the desired interval.
5. Try using ______________to change the form of the equation. It may be helpful to square each side of the
equation first. In this case, check for ______________solutions.
Equations with Half-Angles
EXAMPLE 6 Solving an Equation with a Half-Angle
Solve the equation 2sin 12
x
(a) over the interval 0, 2 , and (b) for all solutions.
Equations with Multiple Angles
EXAMPLE 7 Solving an Equation Using a Double Angle
Solve cos(2𝑥) =√2
2 over the interval 0, 2 .
EXAMPLE 8 Solving an Equation Using a Multiple-Angle
Solve sin(3𝑥) =−1
2
(a) over the interval [0°, 360°), and (b) for all solutions.
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